We introduce an exact, two-parameter family of static, spherically-symmetric, constant-curvature $Λ$-vacuum solutions within the four-dimensional Starobinsky $f(R)=R+αR^2+2Λ$ model. When the bare cosmological constant is precisely fine-tuned to $Λ= 1/(8α)$, the scalar curvature is rigidly fixed such that the derivative $f'(R)=1+2αR$ identically vanishes. Because this derivative acts as the effective multiplier for the standard curvature terms in the modified field equations, its global vanishing mathematically erases the normal rules of gravitational dynamics, demonstrating that the family represents a pathological boundary to the space of viable physical geometries. This exact decoupling of the field equations permits the existence of a fundamentally unconstrained $1/r^2$ integration constant in the metric, which functions as a purely geometric Reissner-Nordstrom hair mimicker. However, any infinitesimal classical deviation from this exact boundary instantaneously destroys the degeneracy, rigorously forcing the geometric hair to vanish and discontinuously collapsing the spacetime back into the standard, dynamical Schwarzschild-de Sitter solution. We provide the exact derivation of this spacetime and methodically highlight its physical pathologies, including the identically vanishing Wald entropy of the associated black hole horizons, the strict divergence of the effective gravitational coupling, the complete breakdown of the test-particle approximation, and the onset of severe ghost instabilities. Ultimately, this exact solution functions as a ``do not enter'' sign within the Starobinsky model, pedagogically illustrating the extreme fragility and physical hostility of degenerate, purely mathematical solutions in highly non-linear $f(R)$ gravity theories.
Circular stable orbits in $f(R)$ realistic static and spherically-symmetric spacetimes
Universidad Complutense de Madrid（马德里康普顿斯大学）；Universidad de Salamanca（萨拉曼卡大学）；University of Cape Town（开普敦大学）；Instituto de Física Corpuscular (IFIC), CSIC‐Universitat de València（瓦伦西亚大学与西班牙国家研究委员会联合粒子物理研究所）；Universidade Federal do Ceará（塞阿拉联邦大学）
C-infinity Compact-Support Wormholes with Exact Schwarzschild Exterior in Trace-Coupled Gravity
National University of Computer and Emerging Sciences（国立计算机与新兴科学大学）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；School of Computing and Mathematical Sciences, University of Leicester（莱斯特大学计算与数学科学学院）；Department of Physics, College of Sciences, Princess Nourah bint Abdulrahman University（公主努拉·宾特·阿卜杜勒拉赫曼大学理学院物理学系）
